TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE18-BOTT-COST-REVIEW.txt Original file SHA-256: bf5fc5c50a88338917e40f37b0ec279c42a09add355adf2051c7e8c703a224b6 Inclusive original lines: 1-104 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | 000004 | Original: documents/archive--paper-latex--tfpt-42/sections/ 000005 | 012-subsection-bosonic-relative-index-and-higgs-selection.tex:84-94. 000006 | SHA256 8140f7618b51292b8acf79fdc999cac9482842626ed07db4edb54dc91c9499d6. 000007 | The displayed old asymptotic functional is 000008 | tau2 |wind det Phi2| + tau3 |wind det Phi3| + m3^2 dim ker B_E3,+. 000009 | The remark explicitly assigns structural weight to the determinant-class 000010 | theorem, not this older energetic intuition. No loop Dirichlet energy or 000011 | pointwise metric on U(2) is specified in that formula. 000012 | 000013 | [OMITTED: non-scientific conversation or execution metadata] 000014 | [OMITTED: non-scientific conversation or execution metadata] 000015 | [OMITTED: non-scientific conversation or execution metadata] 000016 | [OMITTED: non-scientific conversation or execution metadata] 000017 | [OMITTED: non-scientific conversation or execution metadata] 000018 | [OMITTED: non-scientific conversation or execution metadata] 000019 | [OMITTED: non-scientific conversation or execution metadata] 000020 | [OMITTED: non-scientific conversation or execution metadata] 000021 | [OMITTED: non-scientific conversation or execution metadata] 000022 | [OMITTED: non-scientific conversation or execution metadata] 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | [OMITTED: non-scientific conversation or execution metadata] 000025 | [OMITTED: non-scientific conversation or execution metadata] 000026 | [OMITTED: non-scientific conversation or execution metadata] 000027 | [OMITTED: non-scientific conversation or execution metadata] 000028 | [OMITTED: non-scientific conversation or execution metadata] 000029 | [OMITTED: non-scientific conversation or execution metadata] 000030 | [OMITTED: non-scientific conversation or execution metadata] 000031 | [OMITTED: non-scientific conversation or execution metadata] 000032 | [OMITTED: non-scientific conversation or execution metadata] 000033 | 000034 | [OMITTED: non-scientific conversation or execution metadata] 000035 | [OMITTED: non-scientific conversation or execution metadata] 000036 | [OMITTED: non-scientific conversation or execution metadata] 000037 | [OMITTED: non-scientific conversation or execution metadata] 000038 | [OMITTED: non-scientific conversation or execution metadata] 000039 | [OMITTED: non-scientific conversation or execution metadata] 000040 | [OMITTED: non-scientific conversation or execution metadata] 000041 | [OMITTED: non-scientific conversation or execution metadata] 000042 | [OMITTED: non-scientific conversation or execution metadata] 000043 | [OMITTED: non-scientific conversation or execution metadata] 000044 | [OMITTED: non-scientific conversation or execution metadata] 000045 | [OMITTED: non-scientific conversation or execution metadata] 000046 | [OMITTED: non-scientific conversation or execution metadata] 000047 | [OMITTED: non-scientific conversation or execution metadata] 000048 | 000049 | There is also an explicit gauge test. The ratio g_epsilon g_0^-1 is the 000050 | boundary value of the smooth disk gauge diag(exp(i epsilon y),1). Thus 000051 | the two clutches, with their fully transported connections, define gauge 000052 | isomorphic bundle data. The displayed loop Dirichlet energy changes under 000053 | this allowed change of hemisphere frame. It is not a gauge-invariant 000054 | source functional unless additional physical framing rules are supplied. 000055 | 000056 | 3. Framing and the retained positive construction 000057 | The rank1 projector family and its normal family index remain valid under 000058 | their already stated premises. The same projector FORM was present in 000059 | the September30 Bott source; see BOTT-MAP-DEDUP.txt for different variable 000060 | roles. A constant diagonal gauge transformation conjugates P. Whether a 000061 | relative orientation is observable depends on the full gauged source and 000062 | other fields; merely saying one hemisphere is framed is neither a full 000063 | physical selection theorem nor a proof of source uniqueness. 000064 | In fact one fixed hemisphere alone is insufficient under the usual full 000065 | smooth gauge group on the other disk. For any rank1 P,Q let 000066 | g_P=I+(z-1)P, g_Q=I+(z-1)Q, b=g_Q^-1 g_P. 000067 | Then det b=1, so b:S1->SU2 extends over the other disk, and 000068 | (a=I) g_P b^-1=g_Q. 000069 | All these projectors remain equivalent while the first hemisphere frame 000070 | is fixed. A boundary restriction, a fixed connection or a physical 000071 | relative field is needed to obstruct this change, and must be derived. 000072 | Original operational D_coll is itself part of the supplied seed. The 000073 | collar-completion theorem derives B_Sigma from that operator, and does not 000074 | specify a unique weak projector from the determinant class alone. 000075 | The older ADMISSIBLE-INDEX-CARRIER-SELECTOR.txt:146-182 already distinguishes 000076 | the degree-one class from an actual common multiplication operator on the 000077 | same source Hilbert space. Preserve that conclusion and its later Spin 000078 | and determinant-line corrections. 000079 | 000080 | 4. First exact operator evaluation on the genuine pair reader 000081 | For the closed circle take a supplied selfadjoint elliptic D0 with scalar 000082 | internal principal symbol and its actual twisted H1 domain. The loop g 000083 | acts on that internal factor. A smooth based unitary loop g has g(0)=g(2pi)=I, 000084 | so M_g maps that domain to itself, including a fixed endpoint twist. 000085 | Set D_g=M_g D0 M_g^*. It has bounded zeroth-order difference from D0 and 000086 | exp(-t D_g^2)=M_g exp(-tD0^2) M_g^*. 000087 | Therefore Tr(exp(-tD_g^2)-exp(-tD0^2))=0 for every t>0, for g_epsilon as 000088 | well as g_0, despite their different Dirichlet energies. This is a direct 000089 | whole-operator calculation, not a spectral cutoff or a choice of vacuum. 000090 | It shows that this particular genuine pair trace alone does not supply 000091 | the new loop energy or select P. It does not settle a P_adm-compressed, 000092 | fixed-marked or inserted source, nor the actual original reflection-pair 000093 | prescription. Nonflat normal curvature is another input. Those actual 000094 | source data must be tested, rather than silently replacing the reader. 000095 | 000096 | 5. Verdict and continuation 000097 | Established: exact HS loop-energy bound and complete equality family. 000098 | Excluded: original topological cost alone selects that equality family. 000099 | Preserved: the global conditional normal/weak orbit family, canonical U2 000100 | dual-evaluation diagram, original joint lift and native clock conjugacy. 000101 | Next source step: compute the actual fixed-domain source functional on 000102 | g_epsilon and g_0, retaining connections, reference and insertions. A 000103 | geometric c1 equality or a newly imposed Dirichlet action cannot replace 000104 | that source evaluation. No all-P1 impossibility, no physical completion.